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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmcoss | Structured version Visualization version GIF version |
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) |
Ref | Expression |
---|---|
eldmcoss | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss3 36847 | . . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅 | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅) |
3 | eldmcnv 36738 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
4 | 2, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1782 ∈ wcel 2107 class class class wbr 5104 ◡ccnv 5631 dom cdm 5632 ≀ ccoss 36566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-coss 36805 |
This theorem is referenced by: eldmcoss2 36853 eldm1cossres 36854 |
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